Random conical tessellations

Daniel Hug∗ and Rolf Schneider

Abstract

We consider tessellations of the Euclidean (d− 1)-sphere by (d− 2)-dimensional greatsubspheres or, equivalently, tessellations of Euclidean d-space by hyperplanes throughthe origin; these we call conical tessellations. For random polyhedral cones defined astypical cones in a conical tessellation by random hyperplanes, and for their dual cones,we study expectations of certain geometric functionals and extend formulas of Coverand Efron (1967), by including additional geometric functionals, among them the conicalintrinsic volumes. For isotropic conical tessellations, we find counterparts to results ofMiles (1961); they can be interpreted as giving the covariance matrix for a random vectorwhose components are total k-face contents of certain spherical random polytopes.

Key words and phrases: Conical tessellation; spherical tessellation; random polyhedralcones; conical quermassintegrals; conical intrinsic volumes; number of k-faces; first andsecond order moments

AMS 2000 subject classifications. 60D05, 52A22; secondary 52A55; 52C35; 52B05

1 Introduction

Random mosaics in Euclidean spaces are an intensively studied topic of stochastic geometry.We refer the reader to Chapter 10 in the book [22] and to the more recent survey articles[3], [9], [23] and [19]. A much investigated particular class, besides the Voronoi tessellations,are hyperplane tessellations, in particular those generated by stationary Poisson processes ofhyperplanes, initiated by the seminal work of Miles [13], [14], [15], [17] and Matheron [10], [11].Relatively little has been done on random tessellations of spaces other than the Euclidean.Tessellations of the sphere of arbitrary dimensions by great subspheres (of codimension 1)were briefly considered by Cover and Efron [4], and those of the two-dimensional sphere inmore detail by Miles [18]. Relations between various densities of random mosaics in sphericalspaces were studied by Arbeiter and Zahle [2].

In the following, we start a more detailed investigation of tessellations of the sphere Sd−1by random great subspheres. Equivalently, we consider random hyperplanes through theorigin in Rd and the tessellations of Rd into convex polyhedral cones which they generate.For that reason, we talk of ‘conical tessellations’.

Our starting point are some older results due to Miles [13] and to Cover and Efron [4],to which we seek conical analogues or extensions, respectively. In his thesis [13], Chap. 11,Miles has obtained results on first and second order moments for some geometric functionalsof typical cells of stationary Poisson hyperplane tessellations in Euclidean spaces (the resultsare surveyed, without proofs, in [17]). The functionals of polytopes considered by him are

∗Supported in part by DFG grants FOR 1548 and HU 1874/4-2.

1

numbers of k-faces, volumes of k-skeletons, and intrinsic volumes. For certain sphericalrandom polytopes, the expected numbers of k-faces and expected volumes of k-skeletonswere studied by Cover and Efron [4].

The spherical counterparts of the intrinsic volumes (also called conical intrinsic volumes)have recently found interesting applications, for example in [1], [12]. Therefore, it seemsappropriate now to pay increased attention to them also in stochastic geometry. Below,we first supplement the results of Cover and Efron [4] on spherical random polytopes, orrandom cones, by also regarding conical intrinsic volumes (Theorem 4.1). Then we proveconical analogues to the mentioned results of Miles [13], in particular on second moments ofvolumes of skeletons (Theorem 8.1). This comprises also the higher-dimensional extension ofthe results of Miles [18], Theorem 6.3, on tessellations of the two-dimensional sphere.

In Section 2 we introduce the geometric functionals of polyhedral cones that will bestudied, and in Section 3 the two types of random cones for which we investigate expectationsor variances of these functionals. Expectation results of Cover and Efron are extended inSection 4. Sections 5 to 7 are then preparatory to the results on second moments, which arefinally obtained in Section 8.

2 Geometric functionals of convex cones

We work in d-dimensional Euclidean space Rd (d ≥ 2), with scalar product 〈· , ·〉, and denoteby Sd−1 its unit sphere. Let σm, m ∈ N0, be the m-dimensional spherical Lebesgue measure(i.e., the m-dimensional Hausdorff measure) on m-dimensional great subspheres of Sd−1. Forn ∈ N we put

ωn := σn−1(Sn−1) =2πn/2

Γ(n/2).

Let Cd denote the set of closed convex cones in Rd, which includes k-dimensional linearsubspaces, k ∈ 0, . . . , d. We equip Cd with the topology induced by the Fell topology (see[22, Sec. 12.2]). A cone C ∈ Cd is called pointed if it does not contain a line. We write PCdfor the set of polyhedral cones in Cd. This set is a Borel subset of Cd. For C ∈ PCd and fork ∈ 0, . . . , d, we denote by Fk(C) the set of k-dimensional faces of C.

For C ∈ Cd, the dual cone is defined by

C := y ∈ Rd : 〈y, x〉 ≤ 0 for all x ∈ C.

This is again a cone in Cd, and C := (C) = C. If C is pointed and d-dimensional, thenC has the same properties. If C ∈ PCd and F ∈ Fk(C) for k ∈ 0, . . . , d, then the normalcone N(C,F ) of C at F is a (d− k)-face of the polyhedral cone C, also called the conjugateface (of F with respect to C) and denoted by FC . If FC = G, then GC = F .

The following fact is occasionally useful.

Lemma 2.1. Suppose that C ∈ Cd is pointed, and let L ⊂ Rd be a linear subspace. Then

L ∩ C 6= 0 ⇔ L⊥ ∩ intC = ∅.

Proof. Suppose that L ∩ C 6= 0. Choose v ∈ L ∩ C, v 6= 0. Suppose there exists y ∈L⊥ ∩ intC. Since y ∈ L⊥, we have 〈y, v〉 = 0. Since y ∈ intC, the points y′ in someneighbourhood of y belong to C and hence satisfy 〈y′, v〉 ≤ 0. But since 〈y, v〉 = 0 andv 6= 0, this is impossible.

2

Suppose that L⊥ ∩ intC = ∅. The disjoint convex sets L⊥ and intC can be separatedby a hyperplane, hence there is a vector v 6= 0 with 〈v, y〉 ≤ 0 for all y ∈ intC and 〈v, z〉 ≥ 0for all z ∈ L⊥; the latter implies 〈v, z〉 = 0 for z ∈ L⊥ and thus v ∈ L. Since C does notcontain a line, intC 6= ∅, hence 〈v, y〉 ≤ 0 holds for all y ∈ C. Therefore, v ∈ C = C.Thus, v ∈ L ∩ C.

A set M ⊂ Sd−1 is spherically convex if posM is convex; here pos denotes the positivehull. To include some degenerate cases in the following, we define pos ∅ := 0. If C ∈ Cd, theset K = C ∩ Sd−1 is called a convex body in Sd−1, and we have C = posK. In particular, theempty set and k-dimensional great subspheres, that is, intersections of (k + 1)-dimensionallinear subspaces with Sd−1, for k ∈ 0, . . . , d−1 (and thus including Sd−1), are convex bodiesin Sd−1. The set of convex bodies in Sd−1 is denoted by Ks (this notation, as well as the term‘convex body’, differ from the usage in [22, Sec. 6.5], where the empty set is excluded). ForK ∈ Ks, the dual convex body K is defined by

K := y ∈ Sd−1 : 〈y, x〉 ≤ 0 for all x ∈ K = (posK) ∩ Sd−1.

To introduce the conical quermassintegrals and the conical intrinsic volumes, we makeuse of the correspondence between convex cones in Rd and spherically convex sets in Sd−1.For the latter, the functionals to be considered were already introduced by Santalo [20, PartIV], with different notation. We follow here the approach of Glasauer [7] and refer to [22,Sec. 6.5] for further details.

Let G(d, k) denote the Grassmannian of k-dimensional linear subspaces of Rd, and let νkbe its normalized Haar measure (the unique rotation invariant Borel probability measure onG(d, k)), k = 0, . . . , d. For K ∈ Ks, the spherical quermassintegrals are defined by

Uj(K) :=1

2

∫G(d,d−j)

χ(K ∩ L) νd−j(dL), j = 0, . . . , d, (1)

where χ denotes the Euler characteristic. (Of course, U0(K) = 12χ(K) and Ud(K) = 0, but

this is included for formal reasons). We recall from [22, p. 262] that if K is a convex bodyin Sd−1 and not a great subsphere, then χ(K ∩ L) = 1K ∩ L 6= ∅ for νd−j almost allL ∈ G(d, d− j). Hence, in this case 2U(K) is the total invariant probability measure of theset of all (d− j)-dimensional linear subspaces hitting K. Since χ(Sk) = 1 + (−1)k for a greatsubsphere Sk of dimension k ∈ 0, . . . , d− 1, we have

Uj(Sk) =

1, if k − j ≥ 0 and even,

0, if k − j < 0 or odd.

For cones C ∈ Cd, we now define

Uj(C) := Uj(C ∩ Sd−1). (2)

If C ∈ Cd is not a linear subspace, then

Uj(C) =1

2

∫G(d,d−j)

1C ∩ L 6= 0 νd−j(dL), j = 0, . . . , d. (3)

If Lk ⊂ Rd is a linear subspace of dimension k, then

Uj(Lk) =

1, if k − j > 0 and odd,

0, if k − j ≤ 0 or even.(4)

3

Let 0 ≤ j ≤ m ≤ d − 1, let M ⊂ Rd be an m-dimensional linear subspace and C ∈ Cd acone with C ⊂M . The image measure of νd−j under the map L 7→ L∩M from G(d, d− j) tothe Grassmannian of (m − j)-subspaces in M is the normalized Haar measure on the latterspace. Here (and subsequently) we tacitly use the fact that νd−j(L ∈ G(d, d− j) : L ∩M /∈G(d,m − j)) = 0; see [22, Lemma 13.2.1]. Therefore, it follows from (1), (2) that Uj(C)does not depend on whether it is computed in Rd or in M .

In particular, for C ∈ Cd and m ∈ 1, . . . , d, we have

dimC ≤ m ⇒ Um−1(C) =σm−1(C ∩ Sd−1)

ωm.

If C ⊂ Cd is not a linear subspace, the duality relation

Uj(C) + Ud−j(C) =

1

2(5)

holds for j = 0, . . . , d. If C is pointed and d-dimensional, this follows from (3) and Lemma2.1. If C ∈ Cd is not a subspace, the assertion can be obtained from the previous case byapproximation, using easily established continuity properties. If C is a subspace, duality isof little interest, in view of (4).

We now recall the spherical intrinsic volumes and refer to [22, Sec. 6.5] for details. Letds be the spherical distance on Sd−1; thus, for x, y ∈ Sd−1, ds(x, y) = arccos 〈x, y〉. ForK ∈ Ks \ ∅ and x ∈ Sd−1, the distance of x from K is ds(K,x) := minds(y, x) : y ∈ K.For 0 < ε < π/2, the (outer) parallel set of K at distance ε is defined by

Mε(K) := x ∈ Sd−1 : 0 < ds(K,x) ≤ ε.

By the spherical Steiner formula, the measure of this set can be written in the form

σd−1(Mε(K)) =d−2∑m=0

gd,m(ε)vm(K)

with

gd,m(ε) := ωm+1ωd−m−1

∫ ε

0cosm ϕ sind−m−2 ϕdϕ

for 0 ≤ ε < π/2. This defines the numbers v0(K), . . . , vd−2(K) uniquely. The definition issupplemented by setting vm(∅) := 0,

vd−1(K) :=σd−1(K)

ωd,

andv−1(K) := vd−1(K

).

Note that vm(Sd−1) = 0 for m = 0, . . . , d − 2 and vd−1(Sd−1) = 1. The numbers vi(K) arethe spherical intrinsic volumes of K. In particular, for K ∈ Ks and m = 0, . . . , d− 1,

dimK ≤ m ⇒ vm(K) =σm(K)

ωm+1.

For spherical polytopes, the spherical intrinsic volumes have representations in terms ofangles, similar as in the Euclidean case. For a spherical polytope P and for k ∈ 0, . . . , d−2,

4

we denote by Fk(P ) the set of k-faces of P . Let P be a spherical polytope and F ∈ Fk(P ).The external angle γ(F, P ) of P at F is defined by

γ(F, P ) := γ(posF,posP ) :=σd−k−2(N(posP,posF ) ∩ Sd−1)

ωd−k−1.

With these notations, we have

vm(P ) =1

ωm+1

∑F∈Fm(P )

σm(F )γ(F, P ), m = 0, . . . , d− 2.

For cones C ∈ Cd, the conical intrinsic volumes are now defined by

Vm(C) := vm−1(C ∩ Sd−1), m = 0, . . . , d.

The shift in the index has the advantage that the highest occurring index is equal to themaximal possible dimension of C. Since C is a cone, there is no danger of confusion with theintrinsic volumes of compact convex bodies.

For a cone C ∈ Cd with dimC = k, the internal angle of C at 0 is defined by

β(0, C) =σk−1(C ∩ Sd−1)

ωk.

Then, for an arbitrary polyhedral cone C ∈ PCd and for m = 1, . . . , d− 1, we have

Vm(C) =∑

F∈Fm(C)

β(0, F )γ(F,C).

In particular, if dimC = m, then Vm(C) = β(0, C).

In contrast to the quermassintegrals and intrinsic volumes of convex bodies in Euclideanspace, which differ only by their normalizations, the conical quermassintegrals and conicalintrinsic volumes are essentially different functionals. However, they are closely related. Aspherical integral-geometric formula of Crofton type (see [22, (6.63)]) implies that

Uj(C) =

b d−1−j2c∑

k=0

Vj+2k+1(C) (6)

for C ∈ Cd and j = 0, . . . , d− 1. From (6), it follows that

Vj = Uj−1 − Uj+1 for j = 1, . . . , d− 2,Vd−1 = Ud−2,Vd = Ud−1.

(7)

The duality relationVm(C) = Vd−m(C), m = 0, . . . , d (8)

holds for C ∈ Cd. For m ∈ 0, d it holds by definition. For m ∈ 1, . . . , d − 1, it followsfrom (5) and (7) if C is not a subspace, and from

Vj(Lk) = δjk (9)

(Kronecker symbol) if C = Lk is a k-dimensional subspace; here (9) follows from (4) and (7).

5

As did Miles [13, Sec. 5.8] for convex polytopes in Rd, we use the conical quermassintegralsto define a more general series of functionals for polyhedral cones, which comprises someinteresting special cases. For C ∈ PCd, k = 1, . . . , d and j = 0, . . . , k − 1, let

Yk,j(C) :=∑

F∈Fk(C)

Uj(F ). (10)

Then, in particular,dimC = d ⇒ Yd,j(C) = Uj(C).

If C ∈ PCd is such that the k-faces of C are not linear subspaces, then

Yk,0(C) =1

2fk(C), (11)

where fk(C) denotes the number of k-faces of C.

Further, for C ∈ PCd and k ∈ 1, . . . , d, we define the functional Λk by

Λk(C) :=∑

F∈Fk(C)

Vk(F ). (12)

Since the conical intrinsic volumes and the conical quermassintegrals are intrinsicallydefined, it follows from (7) that

Yk,k−1(C) = Λk(C).

3 Conical tessellations and the Cover–Efron model

In this section, we introduce random conical tessellations and the two basic types of randompolyhedral cones that they induce. These random cones were first considered by Coverand Efron [4]. We slightly modify and formalize the approach of [4], to meet our laterrequirements.

Recall that G(d, d− 1) denotes the Grassmannian of (d− 1)-dimensional linear subspacesof Rd. We say that hyperplanes H1, . . . ,Hn ∈ G(d, d− 1) are in general position if any k ≤ dof them have an intersection of dimension d− k. For a vector x ∈ Rd \ 0, let

x⊥ = y ∈ Rd : 〈y, x〉 = 0, x− = y ∈ Rd : 〈y, x〉 ≤ 0.

We shall repeatedly make use of the duality

(posx1, . . . , xn) =

n⋂i=1

x−i , posx1, . . . , xn =

(n⋂i=1

x−i

)(13)

for x1, . . . , xn ∈ Rd.Vectors x1, . . . , xn ∈ Rd are said to be in general position if any d or fewer of these vectors

are linearly independent. Thus, the hyperplanes x⊥1 , . . . , x⊥n are in general position if and

only if x1, . . . , xn are in general position. If this is the case, then

posx1, . . . , xn 6= Rd ⇔n⋂i=1

x−i 6= 0 ⇔ dimn⋂i=1

x−i = d, (14)

6

where the last implication ⇒ follows from general position. In fact, suppose that C :=⋂ni=1 x

−i satisfies 0 < k = dimC < d. Let Lk = linC. Choose p ∈ relintC and define

I := i ∈ 1, . . . , n : p ∈ x⊥i , hence p ∈ intx−j for j ∈ 1, . . . , n \ I. Then C ⊂⋂i∈I x

⊥i

implies that Lk ⊂⋂i∈I x

⊥i ⊂

⋂i∈I x

−i . Since p ∈ intx−j for j ∈ 1, . . . , n \ I, we also have⋂

i∈I x−i ⊂ Lk, and thus Lk =

⋂i∈I x

⊥i =

⋂i∈I x

−i and L⊥k = posxi : i ∈ I, by (13). But

then necessarily |I| ≥ d − k. The assumption of general position implies that |I| = d − k,which is a contradiction to L⊥k = posxi : i ∈ I.

Let H1, . . . ,Hn ∈ G(d, d− 1) be in general position. The hyperplanes H1, . . . ,Hn inducea tessellation T of Rd into d-dimensional polyhedral cones. We call T a conical tessellation ofRd. For k ∈ 1, . . . , d, the set of k-faces of T is defined as the union of the sets of k-faces ofthese polyhedral cones (the d-dimensional cones are the d-faces). We write Fk(H1, . . . ,Hn)for the set of k-faces of the tessellation T . Later, we shall often abbreviate (H1, . . . ,Hn) =: ηnand then write Fk(ηn) for Fk(H1, . . . ,Hn). By fk(T ) we denote the number of k-faces of thetessellation T .

The spherical polytopes C ∩ Sd−1, where C is a cone of T , form a tessellation of thesphere Sd−1, or spherical tessellation. In the following, it will be more convenient to workwith convex cones than with their intersections with Sd−1.

If we denote by H− one of the two closed halfspaces bounded by the hyperplane H, thenit follows from (14) that the d-dimensional cones of the tessellation T induced by H1, . . . ,Hn

are precisely the cones different from 0 of the form

n⋂i=1

εiH−i , εi = ±1.

We call these cones the Schlafli cones induced by H1, . . . ,Hn, n ≥ 1, because Schlafli (gener-alizing a result of Steiner) has shown that there are exactly

C(n, d) := 2d−1∑r=0

(n− 1

r

)(15)

of them (the simple inductive proof is reproduced in [22, Lem. 8.2.1]; also references are foundthere). We consistently define C(0, d) := 1 (where the only cone is Rd itself) and C(n, d) := 0for n < 0.

Each choice of d−k indices 1 ≤ i1 < · · · < id−k ≤ n determines a k-dimensional subspaceL = Hi1 ∩ · · · ∩ Hid−k . For i ∈ 1, . . . , n \ i1, . . . , id−k, the intersections of L with thehyperplanes Hi are in general position in L and hence determine C(n − d + k, k) Schlaflicones with respect to L. Each of these is a k-face of the tessellation T , and each k-face of Tis obtained in this way. Thus, the total number of k-faces is given by

fk(T ) =

(n

d− k

)C(n− d+ k, k) =: C(n, d, k), (16)

for k = 1, . . . , d. In particular, fk(T ) = 1 if n = d− k and fk(T ) = 0 if n < d− k.

Now we turn to random cones. The random vectors appearing in the following can beassumed as unit vectors, since only their spanned rays are relevant. All measures on Sd−1or G(d, d− 1) appearing in the following are Borel measures. Generally, we denote by B(T )the σ-algebra of Borel sets of a given topological space T . Let φ be a probability measureon Sd−1 which is symmetric with respect to 0 (also called even) and assigns measure zero to

7

each (d− 2)-dimensional great subsphere. Let X1, . . . , Xn be independent random points inSd−1 with distribution φ. With probability 1, they are in general position. In the following,we denote probabilities by P and expectations by E.

From Schlafli’s result (15), Wendel has deduced that

p(d)n := P(posX1, . . . , Xn 6= Rd) =C(n, d)

2n(17)

(see [22, Thm. 8.2.1]). This result, having an essentially geometric core, does not depend onthe choice of the distribution φ, as long as the latter has the specified properties.

Cover and Efron [4] have considered the spherically convex hull of X1, . . . , Xn, under thecondition that this convex hull is different from the whole sphere. We talk of the Cover–Efronmodel if a spherically convex random polytope or its spanned cone is generated in this way.

Definition 3.1. Let φ be as above. Let n ∈ N and let X1, . . . , Xn be independent randompoints with distribution φ. The

(φ, n)-Cover–Efron cone Cn

is the random cone defined as the positive hull of X1, . . . , Xn under the condition that this isdifferent from Rd.

Thus, Cn is a random convex cone with distribution given by P(Cn = Rd) = 0 and

P(Cn ∈ B) =1

p(d)n

∫(Sd−1)n

1B(posx1, . . . , xn)φn(d(x1, . . . , xn)) (18)

for B ∈ B(PCdp), where PCdp := PCd \ Rd. Hence C ∈ B ⊂ PCdp implies C 6= Rd.By duality, the Cover–Efron model is connected to random conical tessellations, as we

now explain.

Let φ∗ be the image measure of φ under the mapping x 7→ x⊥ from the sphere Sd−1 to theGrassmannian G(d, d− 1). Every probability measure φ∗ on G(d, d− 1) that assigns measurezero to each set of hyperplanes in G(d, d− 1) containing a fixed line is obtained in this way.Let H1, . . . ,Hn be independent random hyperplanes in G(d, d−1) with distribution φ∗. Withprobability 1, they are in general position.

Definition 3.2. Let φ∗ be as above. Let n ∈ N and let H1, . . . ,Hn be independent randomhyperplanes with distribution φ∗. The

(φ∗, n)-Schlafli cone Sn

is obtained by picking at random (with equal chances) one of the Schlafli cones induced byH1, . . . ,Hn.

Since consecutive random constructions, of which this is an example, will also appear later,we indicate, once and for all, how such a procedure can be formalized. Let Ωn

1 := G(d, d−1)n∗be the set of n-tuples of (d − 1)-subspaces in general position. The probability measure Pnon Ωn

1 is defined by Pn := φ∗n Ωn1 (where denotes the restriction of a measure). We

interpret the choice described in Definition 3.2 as a two-step experiment and define a kernelK1

2 : Ωn1 × B(PCd)→ [0, 1] by

K12 (ηn, B) :=

1

C(n, d)

∑C∈Fd(ηn)

1B(C)

8

for ηn ∈ Ωn1 and B ∈ B(PCd). Then (following, e.g., [6, Satz 1.8.10]), we define a probability

measure Pn ×K12 on B(Ωn

1 )⊗ B(PCd) by

(Pn ×K12 )(A) =

∫G(d,d−1)n

∫PCd

1A(ηn, ω2)K12 (ηn, dω2)φ

∗n(dηn)

=

∫G(d,d−1)n

1

C(n, d)

∑C∈Fd(ηn)

1A(ηn, C)φ∗n(dηn)

for A ∈ B(Ωn1 )⊗ B(PCd). Now Sn is defined as the random cone whose distribution is equal

to (Pn ×K12 )(Ωn

1 × ·). Thus,

P(Sn ∈ B) =

∫G(d,d−1)n

1

C(n, d)

∑C∈Fd(H1,...,Hn)

1B(C)φ∗n(d(H1, . . . ,Hn)) (19)

for B ∈ B(PCd).To relate Sn and Cn, we rewrite equation (18), using the symmetry of φ and then (17)

and (13). For B ∈ B(PCdp), we obtain

P(Cn ∈ B) =1

p(d)n

∫(Sd−1)n

1

2n

∑εi=±1

1B(posε1x1, . . . , εnxn)φn(d(x1, . . . , xn))

=

∫(Sd−1)n

1

C(n, d)

∑εi=±1

1B

((n⋂i=1

εix−i

))φn(d(x1, . . . , xn))

=

∫(Sd−1)n

1

C(n, d)

∑C∈Fd(x⊥1 ,...,x⊥n )

1B(C)φn(d(x1, . . . , xn))

=

∫G(d,d−1)n

1

C(n, d)

∑C∈Fd(H1,...,Hn)

1B(C)φ∗n(d(H1, . . . ,Hn))

= P(Sn ∈ B),

where (19) was used in the last step. Thus,

Cn = Sn in distribution, (20)

since also P(Sn = Rd) = P(Sn = ∅) = 0.

4 Expectations for random Schlafli and Cover–Efron cones

In this section, φ∗ is a probability measure on the Grassmannian G(d, d−1) with the propertythat it is zero on each set of hyperplanes containing a fixed line through 0. For n ∈ N, weconsider the (φ∗, n)-Schlafli cone and want to compute the expectations of the geometricfunctionals Yk,j , defined by (10), for this random cone.

In his study of Poisson hyperplane tessellations in Euclidean spaces, Miles [13, Chap. 11]has employed the idea of defining, by means of combinatorial selection procedures, differ-ent weighted random polytopes, which could then be combined to give results about first

9

and second moments. In this and subsequent sections, we adapt this approach to conicaltessellations.

First we describe a combinatorial random choice. Let H1, . . . ,Hn ∈ G(d, d− 1) be hyper-planes in general position, and let L ∈ G(d, k), for k ∈ 1, . . . , d, be a k-dimensional linearsubspace in general position with respect to H1, . . . ,Hn, which means that H1∩L, . . . ,Hn∩Lare (k− 1)-dimensional subspaces of L which are in general position in L. Let j ∈ 1, . . . , k.The tessellation TL induced in L by H1 ∩ L, . . . ,Hn ∩ L, has C(n, k, j) faces of dimension j,by (16). If n < k− j, then clearly C(n, k, j) = 0. The following is an immediate consequenceof general position.

Lemma 4.1. Let j ≥ 1. To each j-face Fj of TL, there is a unique (d− k+ j)-face F of thetessellation T induced by H1, . . . ,Hn, such that Fj = F ∩ L.

Conversely, if F ∈ Fd−k+j(T ) and F ∩ L 6= 0, then F ∩ L is a j-face of TL.

In the following, we assume that n ≥ k− j. We choose one of the j-faces of TL at random(with equal chances) and denote it by Fj . Then Fj = L∩F with a unique face F ∈ Fd−k+j(T ).The face Fj is contained in 2k−j Schlafli cones of TL and thus in 2k−j Schlafli cones of T .These are precisely the Schlafli cones of T that contain F . We select one of these at random(with equal chances) and call it C [k,j](H1, . . . ,Hn, L).

Let H1, . . . ,Hn be independent random hyperplanes with distribution φ∗. We apply thedescribed procedure to these hyperplanes and to a random k-dimensional subspace. Thisrandom subspace will here be chosen as explained below, and in a different way in Section 6.

Let L ∈ G(d, k) be a random subspace with distribution νk, which is independent ofH1, . . . ,Hn; for k = d, L = Rd is deterministic. We may assume, since this happens withprobability 1, that H1, . . . ,Hn and L are in general position. Then we define

C [k,j]n := C [k,j](H1, . . . ,Hn,L). (21)

More formally, C[k,j]n is a random polyhedral cone with distribution given by

P(C [k,j]n ∈ B) (22)

=

∫G(d,d−1)n

∫G(d,k)

1

C(n, k, j)

∑F∈Fd−k+j(ηn)

F∩L6=0

1

2k−j

∑C∈Fd(ηn)C⊃F

1B(C) νk(dL)φ∗n(dηn)

for B ∈ B(PCd) and n ≥ k − j (recall that ηn is a shorthand notation for (H1, . . . ,Hn)).

If n > k − j, then almost surely F ∈ Fd−k+j(ηn) is not a linear subspace. Thus, (3)implies that the inner integral in (22), up to the combinatorial factors, can be written as∫

G(d,k)

∑F∈Fd−k+j(ηn)

1F ∩ L 6= 0∑

C∈Fd(ηn)

1F ⊂ C1B(C) νk(dL)

=∑

C∈Fd(ηn)

1B(C)∑

F∈Fd−k+j(ηn)

1F ⊂ C∫G(d,k)

1F ∩ L 6= 0 νk(dL)

=∑

C∈Fd(ηn)

1B(C)∑

F∈Fd−k+j(ηn)

1F ⊂ C2Ud−k(F )

= 2∑

C∈Fd(ηn)

1B(C)Yd−k+j,d−k(C),

10

according to (10). Therefore, we obtain

P(C [k,j]n ∈ B) =

2

2k−jC(n, k, j)

∫G(d,d−1)n

∑C∈Fd(ηn)

1B(C)Yd−k+j,d−k(C)φ∗n(dηn). (23)

From (19) and (23) (both formulated for expectations) we get, for every nonnegative,measurable function g on PCd and n > k − j, the equation

E g(C [k,j]n ) =

2C(n, d)

2k−jC(n, k, j)E (gYd−k+j,d−k)(Sn). (24)

With g = 1, (24) reads

EYd−k+j,d−k(Sn) =2k−jC(n, k, j)

2C(n, d), (25)

for 1 ≤ j ≤ k ≤ d and n > k − j, so that we can also write

E g(C [k,j]n ) =

E (gYd−k+j,d−k)(Sn)

EYd−k+j,d−k(Sn).

Thus, the distribution of C[k,j]n is obtained from the distribution of Sn by weighting it with

the function Yd−k+j,d−k. This is the conical counterpart to [13, Sec. 11.3, Lemma]. In analogyto [13, Sec. 11.3], we point out some special cases.

If k = j = 1, the procedure is equivalent to choosing a uniform random point in Sd−1,independent of H1, . . . ,Hn, and taking for C

[1,1]n the Schlafli cone containing it. The weight

function satisfies Yd,d−1(C) = Vd(C).

If k = d, the procedure is equivalent to choosing a j-face of the tessellation T at random(with equal chances) and then choosing at random (with equal chances) one of the Schlafli

cones containing it, which gives C[d,j]n . The weight function satisfies Yj,0(C) = 1

2fj(C), sincethe assumption n > d−j implies that the j-faces of C are not linear subspaces. In particular,

for j = d it is constant, and C[d,d]n = Sn in distribution.

By specialization, the equation (25) includes some results obtained by Cover and Efron[4]. They proved for k = 1, . . . , d that

Efk(Sn) =2d−k

(nd−k)C(n− d+ k, k)

C(n, d)

(formula (3.1) in [4], after correction of misprints). This equation is obtained from (25) bychoosing k = d and then replacing j by k (and observing (11) and (16)) for n > d − k. Forn = d− k both sides are equal to 1, and for n < d− k both sides are zero. The duality (20)gives

Efk(Cn) =2k(nk

)C(n− k, d− k)

C(n, d)

for k = 0, . . . , d− 1, which is formula (3.3) in [4].

In the summary of their paper [4], Cover and Efron also announced results on the ‘expectednatural measure of the set of k-faces’. As such a natural measure one can consider thefunctional Λk defined by (12) for polyhedral cones (or its natural analogue in the case ofspherical polytopes). With this definition, one has

EΛk(Sn) =2d−k

(nd−k)

C(n, d), (26)

11

for k = 1, . . . , d, and

EΛk(Cn) =

(nk

)C(n− k, d− k)

C(n, d), (27)

for k = 1, . . . , d − 1. In contrast to (27), relation (26) holds also for k = d, by (25). Coverand Efron did not formulate these results; however, some arguments leading to them arecontained in their Theorems 2 and 4. We note that (26) is the special case of (25) which isobtained by replacing k by d− k + 1 and setting j = 1. Here we use that for n > d− k, thek-faces of Sn are not in G(d, k). For n ≤ d− k, the equation is apparently true as well.

For (27), we extend and complete the arguments given in [4]. For the proof, we canassume that n ≥ k. Let k ∈ 1, . . . , d− 1. By (20) and (19),

EΛk(Cn) = EΛk(Sn) =

∫G(d,d−1)n

1

C(n, d)

∑C∈Fd(ηn)

Λk(C)φ∗n(dηn).

Let ηn = (H1, . . . ,Hn), where H1, . . . ,Hn ∈ G(d, d − 1) are in general position. LetF ∈ Fd−k(ηn). Then there are indices 1 ≤ i1 < · · · < ik ≤ n such that

F ⊂ Li1,...,ik := Hi1 ∩ · · · ∩Hik .

Let CF be the set of Schlafli cones C ∈ Fd(ηn) with F ⊂ C. Let uj be a unit normal vector ofHij , j = 1, . . . , k. Then the cones C ∈ CF are in one-to-one correspondence with the choicesε1, . . . , εk ∈ −1, 1 such that

C ⊂k⋂j=1

εju−j .

The face of C conjugate to F (with respect to C) is then given by

FC = posε1u1, . . . , εkuk.

It follows that the faces FC , C ∈ CF , form a tiling of L⊥i1,...,ik , and therefore∑C∈Fd(ηn)

1F ⊂ CVk(FC) = 1. (28)

The faces F ∈ Fd−k(ηn) with F ⊂ Li1,...,ik are the Schlafli cones of the tessellation inducedin Li1,...,ik , hence there are precisely C(n− k, d− k) of them. Now we obtain, using (28) andthe latter remark,∑

C∈Fd(ηn)

Λk(C) =

∑C∈Fd(ηn)

∑G∈Fk(C)

Vk(G) =∑

C∈Fd(ηn)

∑F∈Fd−k(C)

Vk(FC)

=∑

F∈Fd−k(ηn)

∑C∈Fd(ηn)

1F ⊂ CVk(FC)

=∑

1≤i1<···<ik≤n

∑F∈Fd−k(ηn)

1F ⊂ Li1,...,ik∑

C∈Fd(ηn)

1F ⊂ CVk(FC)

=

(n

k

)C(n− k, d− k),

which yields (27).

12

The following expectations do not appear in [4]. Replacing k and j in (25) both by d− k,then for n ≥ 1 we obtain

EUk(Sn) =C(n, d− k)

2C(n, d), k = 0, . . . , d− 1.

Note that if n ≤ d− k, then both sides of the equation are equal to 1/2.

Since Cn is almost surely pointed, the dualities (5) and (20) yield

EUk(Cn) =C(n, d)− C(n, k)

2C(n, d), k = 1, . . . , d− 1,

where both sides of the equation are equal to 0 if n < k.

The relations given in (7) can now be used to deduce the following.

Theorem 4.1. The expected conical intrinsic volumes of the (φ∗, n)-Schlafli cone Sn and the(φ, n)-Cover–Efron cone Cn are given by

EVj(Sn) =

(n

d− j

)C(n, d)−1

for j = 1, . . . , d and

EVj(Cn) =

(n

j

)C(n, d)−1, j = 1, . . . , d− 1,(

n− 1

d− 1

)C(n, d)−1, j = d.

Of course, for j = 1, . . . , d − 1, the dualities (8) and (20) can also be used to derive thesecond relation from the first.

We point out that the results obtained so far hold for general distributions φ∗, as specifiedat the beginning of this section.

5 Some first and second order moments

For stationary random mosaics in Rd it is known (e.g., [22, Thm. 10.4.1]) that the distributionof the zero cell is, up to translations, the volume-weighted distribution of the typical cell. Inthis section, we derive an analogous statement for conical tessellations (Lemma 5.2). Togetherwith the expectation (33), it yields the second moment (35), which is an essential prerequisitefor the proof of the main result, Theorem 8.1.

Recall that νd−1 denotes the unique rotation invariant probability measure on the Grass-mannianG(d, d−1). The subsequent results require this special distribution for the consideredrandom hyperplanes, instead of the general distribution φ∗ of the previous sections.

First we formulate a simple lemma.

Lemma 5.1. If A ∈ B(Sd−1) and k ∈ 1, . . . , d− 1, then∫G(d,d−1)k

σd−k−1(A ∩H1 ∩ · · · ∩Hk) νkd−1(d(H1, . . . ,Hk)) =

ωd−kωd

σd−1(A). (29)

13

Proof. As a function of A, the left-hand side of (29) is a finite measure, which, due to therotation invariance of νd−1, must be invariant under rotations. Up to a constant factor, thereis only one such measure on B(Sd−1), namely σd−1. The choice A = Sd−1 then reveals thefactor.

Now let H1, . . . ,Hn be independent random hyperplanes through 0 with distributionνd−1. Before treating the (νd−1, n)-Schlafli cone, we consider a different random cone, whichcorresponds to the zero cell in the theory of Euclidean tessellations. Let e ∈ Sd−1 be a fixedvector. With probability 1, the vector e is contained in a unique Schlafli cone induced byH1, . . . ,Hn, and we denote this cone by Sen. If e /∈ H ∈ G(d, d − 1), we denote by He theclosed halfspace bounded by H that contains e.

Let k ∈ 0, . . . , d − 1. Almost surely, each (d − k)-face of Sen is the intersection of Senwith exactly k of the hyperplanes H1, . . . ,Hn. Conversely, each intersection of k distincthyperplanes from H1, . . . ,Hn a.s. intersects Sen either in a (d− k)-face or in 0. Observingthis, we compute

EΛd−k(Sen) = E

∑1≤i1<···<ik≤n

Vd−k(Sen ∩Hi1 ∩ · · · ∩ Hik)

=∑

1≤i1<···<ik≤nEVd−k(He1 ∩ · · · ∩ Hen ∩Hi1 ∩ · · · ∩ Hik)

=

(n

k

)EVd−k(Hek+1 ∩ · · · ∩ Hen ∩H1 ∩ · · · ∩ Hk)

=

(n

k

)∫G(d,d−1)n−k

∫G(d,d−1)k

Vd−k(Hek+1 ∩ · · · ∩He

n ∩H1 ∩ · · · ∩Hk)

× νkd−1(d(H1, . . . Hk)) νn−kd−1 (d(Hk+1, . . . ,Hn)).

If n = k, the outer integration does not appear, and Hek+1 ∩ · · · ∩ Hen has to be interpreted

as Rd. For n < k, both sides of the equation are zero.

By Lemma 5.1, the inner integral is equal to

1

ωdσd−1(H

ek+1 ∩ · · · ∩He

n ∩ Sd−1),

hence we obtain

EΛd−k(Sen) =

(n

k

)EVd(Sen−k) (30)

for k = 0, . . . , d− 1. Here both sides of the equation are zero if n < k, and they are equal to1 for n = k.

We next derive a similar formula for E fd−k(Sen) (in analogy to [21, Sec. 5]). Let k ∈0, . . . , d− 1 and n > k. As above, we obtain

E fd−k(Sen) = E∑

1≤i1<···<ik≤n1Sen ∩Hi1 ∩ · · · ∩ Hik 6= 0

=

(n

k

)∫G(d,d−1)n−k

∫G(d,d−1)k

1Hek+1 ∩ · · · ∩He

n ∩H1 ∩ · · · ∩Hk 6= 0

× νkd−1(d(H1, . . . Hk)) νn−kd−1 (d(Hk+1, . . . ,Hn)).

14

Let G(d, d − 1)k∗ denote the set of all k-tuples of (d − 1)-dimensional linear subspaceswith linearly independent normal vectors. The image measure of νkd−1 under the mapping

(H1, . . . ,Hk) 7→ H1 ∩ · · · ∩Hk from G(d, d − 1)k∗ to G(d, d − k) is the invariant measure νk,hence ∫

G(d,d−1)k1C ∩H1 ∩ · · · ∩Hk 6= 0 νkd−1(d(H1, . . . Hk)) = 2Uk(C)

for C = Hek+1 ∩ · · · ∩ He

n ∈ Cd and νn−kd−1 almost all (Hk+1, . . . ,Hn) ∈ G(d, d − 1)n−k. Weconclude that

E fd−k(Sen) = 2

(n

k

)EUk(Sen−k)

for k ∈ 0, . . . , d− 1 and n > k. If n = k, then E fd−k(Sen) = 1, and the expectation is zerofor n < k.

To compute EVd(Sen), let P ⊂ Sd−1 be a closed spherically convex set containing e.Writing u ∈ Sd−1 in the form u = te+

√1− t2 u with u ∈ e⊥ ∩ Sd−1, we have

σd−1(P ) =

∫e⊥∩Sd−1

∫ 1

cos ρ(P,u)(1− t2)

d−32 dt σd−2(du) (31)

withρ(P, u) = maxρ ∈ [0, π] : (cos ρ)e+ (sin ρ)u ∈ P, u ∈ e⊥ ∩ Sd−1.

Let Zen := Sen∩Sd−1. For fixed u ∈ e⊥∩Sd−1, the distribution function of the random variableρ(Zen, u) is given by

F (x) = P (ρ(Zen, u) < x) = 1−(

1− x

π

)n,

since ρ(Zen, u) > x holds if and only if none of the hyperplanes H1, . . . ,Hn intersects the greatcircular arc connecting e and (cosx)e+ (sinx)u. Let

G(x) :=

∫ 1

cosx(1− t2)

d−32 dt =

∫ x

0sind−2 α dα for x ∈ [0, π].

From (31) we have G(π) = ωd/ωd−1. Since the distribution of the random variable ρ(Zen, u)does not depend on u, we obtain

Eσd−1(Zen) = E∫e⊥∩Sd−1

∫ 1

cos ρ(Zen,u)(1− t2)

d−32 dt σd−2(du)

= ωd−1EG(ρ(Zen, u))

= ωd−1

∫ π

0G(x)F ′(x) dx

= ωd−1

[G(π)−

∫ π

0G′(x)F (x) dx

]

= ωd−1

[ωdωd−1

−∫ π

0sind−2 x

(1−

(1− x

π

)n)dx

]= ωd−1

∫ π

0

(1− x

π

)nsind−2 x dx.

After using the binomial theorem, the integral can be evaluated by using recursion formulasand known definite integrals; e.g., see [8, p. 117]. (The evaluation of the integral for d = 3 in[18, (6.16)] is corrected in [5].)

15

Defining the constant θ(n, d) by

θ(n, d) :=ωd−1ωd

∫ π

0

(1− x

π

)nsind−2 x dx, for n ∈ N0, (32)

and by θ(n, d) := 0 for n < 0, and recalling that Vd(Sen) = σd−1(Z

en)/ωd, we can write the

result asEVd(Sen) = θ(n, d). (33)

Note that θ(0, d) = 1. As a corollary, we obtain from (30) that

EΛd−k(Sen) =

(n

k

)θ(n− k, d) (34)

for k ∈ 0, . . . , d− 1. For n = k both sides are equal to 1, and they are zero for n < k.

The following lemma relates the distribution of Sen to that of the (νd−1, n)-Schlafli coneSn.

Lemma 5.2. Let f be a nonnegative measurable function on PCd which is invariant underrotations. Then

E f(Sen) = C(n, d)E (fVd)(Sn).

Proof. In the following, we denote by ν the invariant probability measure on the rotationgroup SO(d), and we make use of the fact that∫

SO(d)g(ϑe) ν(dϑ) =

1

ωd

∫Sd−1

g(u)σd−1(du)

for every nonnegative measurable function g on Sd−1. Using the rotation invariance of thefunction f and of the probability distribution νd−1, we obtain, with ϑ ∈ SO(d),

E f(Sen) = E∑

C∈Fd(H1,...,Hn)

f(C)1intC(e)

= E∑

C∈Fd(H1,...,Hn)

f(C)1intC(ϑe)

= E∫SO(d)

∑C∈Fd(H1,...,Hn)

f(C)1intC(ϑe) ν(dϑ)

=1

ωdE∫Sd−1

∑C∈Fd(H1,...,Hn)

f(C)1intC(u)σd−1(du)

=1

ωdE

∑C∈Fd(H1,...,Hn)

f(C)σd−1(C ∩ Sd−1)

= C(n, d)E (fVd)(Sn)

by (19) (with φ∗ = νd−1).

From Lemma 5.2 and (34) we get

E (Λd−kVd)(Sn) =

(nk

)θ(n− k, d)

C(n, d)(35)

16

for k = 0, . . . , d− 1. The case k = 0 reads

EV 2d (Sn) =

θ(n, d)

C(n, d).

Equation (35) is a conical counterpart to Miles [13, Thm. 11.1.1]. The special case d = 3of (35) is contained in Miles [18, Thm. 6.3].

6 Another selection procedure

In Section 4, we have used a selection procedure to define a random cone C[k,j]n . This selection

procedure will now be modified. The assumptions are the same as in Section 5: H1, . . . ,Hnare independent random hyperplanes through 0, each with distribution νd−1, the rotationinvariant probability measure on G(d, d− 1).

The second selection procedure is equivalent to a conical analogue of the one in [13,Sec. 11.4], though we describe it in a different way. We assume again that 1 ≤ j ≤ k ≤ d andn ≥ d − j (that is, n − (d − k) ≥ k − j). Now a subspace L ∈ G(d, k) is chosen at random(with equal chances) from the k-dimensional intersections of the hyperplanes H1, . . . ,Hn. (Ifk = d, then L = Rd is deterministic. Corresponding adjustments can be made below.) Thereare indices i1, . . . , id−k ∈ 1, . . . , n such that

L = Hi1 ∩ · · · ∩ Hid−k ,

since n ≥ d− j ≥ d− k. In the following, if ηn = (H1, . . . ,Hn), we denote by ηn〈i1, . . . , id−k〉the (n−d+k)-tuple that remains when Hi1 , . . . ,Hid−k have been removed from (H1, . . . ,Hn).Similarly, Hn〈i1, . . . , id−k〉 is obtained from Hn = (H1, . . . ,Hn). Then, employing the defini-tion (21), we define

D[k,j]n := C [k,j](Hn〈i1, . . . , id−k〉,L).

(Note that the indices i1, . . . , id−k are determined by L.)

Let B ∈ B(PCd). According to the definition of D[k,j]n , we have

P(D[k,j]n ∈ B) =

∫G(d,d−1)n

1(nd−k) ∑

1≤i1<···<id−k≤n

1

C(n− d+ k, k, j)

∑F∈Fd−k+j(ηn〈i1,...,id−k〉)F∩Hi1∩···∩Hid−k 6=0

× 1

2k−j

∑C∈Fd(ηn〈i1,...,id−k〉)

C⊃F

1B(C) νnd−1(dηn).

For k = d, the condition F ∩Hi1∩· · ·∩Hid−k 6= 0 is empty and can be deleted. Moreover, if

n = d− k, then j = k, F = C = Rd and D[k,j]n = D

[k,k]d−k = Rd almost surely. After interchang-

ing the integration and the first summation, the summands of the sum∑

1≤i1<···<id−k≤n are

17

all the same. Therefore, we obtain

P(D[k,j]n ∈ B)

=1

2k−jC(n− d+ k, k, j)

∫G(d,d−1)n

∑F∈Fd−k+j(ηn〈1,...,d−k〉)F∩H1∩···∩Hd−k 6=0

∑C∈Fd(ηn〈1,...,d−k〉)

C⊃F

1B(C) νnd−1(dηn)

=1

2k−jC(n− d+ k, k, j)

∫G(d,d−1)n−d+k

∫G(d,d−1)d−k

∑F∈Fd−k+j(Hd−k+1,...,Hn)

F∩H1∩···∩Hd−k 6=0

×∑

C∈Fd(Hd−k+1,...,Hn)

C⊃F

1B(C) νd−kd−1 (d(H1, . . . ,Hd−k)) νn−d+kd−1 (d(Hd−k+1, . . . ,Hn)). (36)

If n = d−k, then the outer integration is omitted and F = C = Rd. We have split the n-foldintegration, since the image measure of νd−kd−1 under the map (H1, . . . ,Hd−k) 7→ H1∩· · ·∩Hd−kfrom G(d, d − 1)d−k∗ to G(d, k) is (for reasons of rotation invariance) the Haar probabilitymeasure νk on G(d, k). Therefore, for the inner integral we obtain∫

G(d,d−1)d−k

∑F∈Fd−k+j(Hd−k+1,...,Hn)

F∩H1∩···∩Hd−k 6=0

∑C∈Fd(Hd−k+1,...,Hn)

C⊃F

1B(C) νd−kd−1 (d(H1, . . . ,Hd−k))

=

∫G(d,k)

∑F∈Fd−k+j(Hd−k+1,...,Hn)

1F ∩ L 6= 0∑

C∈Fd(Hd−k+1,...,Hn)

C⊃F

1B(C) νk(dL).

Assume now that n > d− j. Then, arguing as in the derivation of (23), we see that the latteris equal to ∑

F∈Fd−k+j(Hd−k+1,...,Hn)

2Ud−k(F )∑

C∈Fd(Hd−k+1,...,Hn)

1C ⊃ F )1B(C)

=∑

C∈Fd(Hd−k+1,...,Hn)

∑F∈Fd−k+j(Hd−k+1,...,Hn)

1C ⊃ F2Ud−k(F )1B(C)

= 2∑

C∈Fd(Hd−k+1,...,Hn)

Yd−k+j,d−k(C)1B(C).

We conclude that

P(D[k,j]n ∈ B) =

2

2k−jC(n− d+ k, k, j)

∫G(d,d−1)n−d+k

×∑

C∈Fd(ηn−d+k)

1B(C)Yd−k+j,d−k(C) νn−d+kd−1 (dηn−d+k).

Together with (19) (for φ∗ = νd−1) this yields

Eg(D[k,j]n ) =

2C(n− d+ k, d)

2k−jC(n− d+ k, k, j)E (gYd−k+j,d−k)(Sn−d+k) (37)

for every nonnegative, measurable function g on PCd and n > d− j. It implies that

E g(D[k,j]n ) =

E (gYd−k+j,d−k)(Sn−d+k)

EYd−k+j,d−k(Sn−d+k).

18

This is the conical counterpart to [13, Thm. 11.5.1] (but in contrast to that, we have noequivalence here: n on the left side and n− d+ k on the right side).

For later application, we note the special case k = j. From (36) and (37) we obtain∫G(d,d−1)n

∑C∈Fd(Hd−j+1,...,Hn)

C∩H1∩···∩Hd−j 6=0

g(C) νnd−1(d(H1, . . . ,Hn)) = C(n− d+ j, j)E g(D[j,j]n )

= 2C(n− d+ j, d)E (gUd−j)(Sn−d+j) (38)

for n > d− j.

7 A geometric identity

To draw conclusions from the previous results, we need a geometric identity, in analogy to[13, Sec. 11.6]. Let ηn = (H1, . . . ,Hn) ∈ G(d, d− 1)n∗ , let j ∈ 1, . . . , d− 1, n > d− j and

Lj := H1 ∩ · · · ∩Hd−j .

Let Fj ∈ Fj(ηn) be a j-face such that Fj ⊂ Lj . Let k ∈ j, . . . , d. We delete the hyperplanesHk−j+1, . . . ,Hd−j . From the tessellation induced by the remaining hyperplanes, we collectthe d-cones containing Fj and then classify their r-faces for fixed r. Thus, we define

Fd(ηn, Fj , k) := C ∈ Fd(ηn〈k − j + 1, . . . , d− j〉) : Fj ⊂ C.

Let r ∈ 1, . . . , d. For p ∈ N with r ≤ p ≤ d and d− p ≤ k − j, let

Fr,p :=F ∈ Fr(C) : C ∈ Fd(ηn, Fj , k), F ⊂ Hi for precisely d− p indices i ∈ 1, . . . , k − j

.

We recall that Λr(C), defined for C ∈ PCd by (12), is the normalized spherical (r − 1)-volume of the (r − 1)-skeleton of C ∩ Sd−1, that is,

Λr(C) =∑

F∈Fr(C)

Vr(F ) =∑

F∈Fr(C)

σr−1(F ∩ Sd−1)ωr

.

We have

∑C∈Fd(ηn,Fj ,k)

Λr(C) =∑

C∈Fd(ηn,Fj ,k)

∑F∈Fr(C)

Vr(F ) =d∑

p=maxr,d−k+j

2d−p∑

F∈Fr,p

Vr(F ),

since each F ∈ Fr,p belongs to precisely 2d−p cones C ∈ Fd(ηn, Fj , k).

Let Q be the unique cone in Fd(ηn〈1, . . . , d− j〉) with Fj ⊂ Q, and define

Cp :=Q ∩Hi1 ∩ · · · ∩Hid−p : 1 ≤ i1 < · · · < id−p ≤ k − j

.

Thus, Cp is a set of p-dimensional cones, and Cd = Q. Each r-face F ∈ Fr,p satisfiesF ⊂ G ∈ Fr(D) for a unique D ∈ Cp and a unique G ∈ Fr(D). Conversely, for D ∈ Cp and

19

G ∈ Fr(D), the r-face G is the union of r-faces from Fr,p, which pairwise have no relativelyinterior points in common. It follows that∑

F∈Fr,p

Vr(F ) =∑D∈Cp

∑F∈Fr(D)

Vr(F ).

We conclude that ∑C∈Fd(ηn,Fj ,k)

Λr(C) =d∑

p=maxr,d−k+j

2d−p∑D∈Cp

∑F∈Fr(D)

Vr(F )

=d∑

p=maxr,d−k+j

2d−p∑D∈Cp

Λr(D). (39)

Relation (39) was derived for any Fj ∈ Fj(ηn) with Fj ⊂ Lj . We sum over all such j-facesand note that C ∈ Fd(ηn〈k − j + 1, . . . , d− j〉) satisfies Fj ⊂ C for some j-face Fj ∈ Fj(ηn)with Fj ⊂ Lj if and only if C ∩ Lj 6= 0. Concerning the set Cp appearing on the right-hand side of (39), we note that Q ∈ Fd(ηn〈1, . . . , d − j〉) satisfies Fj ⊂ Q for some j-faceFj ∈ Fj(ηn) with Fj ⊂ Lj if and only if Q ∩ Lj 6= 0. Therefore, we obtain the geometricidentity ∑

C∈Fd(ηn〈k−j+1,...,d−j〉)C∩Lj 6=0

Λr(C) (40)

=

d∑p=maxr,d−k+j

2d−p∑

1≤i1<···<id−p≤k−j

∑Q∈Fd(ηn〈1,...,d−j〉)

Q∩Lj 6=0

Λr(Q ∩Hi1 ∩ · · · ∩Hid−p),

which will be required in Section 8. (For k = j, the middle sum on the right-hand side hasto be deleted, and the equation becomes a tautology.) This holds for ηn = (H1, . . . ,Hn) ∈G(d, d− 1)n∗ , j ∈ 1, . . . , d− 1, with Lj := H1 ∩ · · · ∩Hd−j , r = 1, . . . , d, k ∈ j, . . . , d andfor n > d− j.

8 A covariance matrix

Now the preceding results are combined, in analogy to [13, Sec. 11.7]. We use (36), extendedto expectations and then applied to the expectation of Λr, for given r ∈ 1, . . . , d. However,it will be convenient to replace the index tuple (1, . . . , d − k) by (k − j + 1, . . . , d − j), forgiven j ∈ 1, . . . , d − 1 and k ∈ j, . . . , d. As before we assume that n > d − j. Then wehave (splitting the multiple integral appropriately)

EΛr(D[k,j]n ) (41)

=1

2k−jC(n− d+ k, k, j)

∫G(d,d−1)n−d+j

∫G(d,d−1)k−j

∫G(d,d−1)d−k

×∑

F∈Fd−k+j(ηn〈k−j+1,...,d−j〉)F∩Hk−j+1∩···∩Hd−j 6=0

∑C∈Fd(ηn〈k−j+1,...,d−j〉)

C⊃F

Λr(C)

× νd−kd−1 (d(Hk−j+1, . . . ,Hd−j)) νk−jd−1(d(H1, . . . ,Hk−j)) ν

n−d+jd−1 (d(Hd−j+1, . . . ,Hn)).

20

(Recall that, for k = d, the condition F ∩ Hk−j+1 ∩ · · · ∩ Hd−j 6= 0 is empty and can bedeleted.) If k > j, we split the first sum above in the form∑

F∈Fd−k+j(ηn〈k−j+1,...,d−j〉)F∩Hk−j+1∩···∩Hd−j 6=0

=∑

1≤i1<···<ik−j≤ni1,...,ik−j /∈k−j+1,...,d−j

∑F∈Fd−k+j(ηn〈k−j+1,...,d−j〉)

F∩Hk−j+1∩···∩Hd−j 6=0, F⊂Hi1∩···∩Hik−j

.

Then, after interchanging in (41) the first summation on the right side and integration, theouter sum has

(n−d+kk−j

)equal terms, hence we obtain (again regrouping the integrals)

EΛr(D[k,j]n ) =

(n−d+kk−j

)2k−jC(n− d+ k, k, j)

∫G(d,d−1)k−j

∫G(d,d−1)n−k+j

×∑

F∈Fd−k+j(ηn〈k−j+1,...,d−j〉)F⊂H1∩···∩Hk−j , F∩Hk−j+1∩···∩Hd−j 6=0

∑C∈Fd(ηn〈k−j+1,...,d−j〉)

C⊃F

Λr(C)

× νn−k+jd−1 (d(Hk−j+1, . . . ,Hn)) νk−jd−1(d(H1, . . . ,Hk−j)).

(If k = j, the condition F ⊂ H1 ∩ · · · ∩ Hk−j is empty and can be deleted.) For fixedH1, . . . ,Hk−j , we consider the inner integral

I :=

∫G(d,d−1)n−k+j

∑F∈Fd−k+j(ηn〈k−j+1,...,d−j〉)

F⊂H1∩···∩Hk−j , F∩Hk−j+1∩···∩Hd−j 6=0

∑C∈Fd(ηn〈k−j+1,...,d−j〉)

C⊃F

Λr(C)

× νn−k+jd−1 (d(Hk−j+1, . . . ,Hn)).

A cone C ∈ Fd(ηn〈k − j + 1, . . . , d − j〉) has a face F ∈ Fd−k+j(ηn〈k − j + 1, . . . , d − j〉)satisfying

F ⊂ H1 ∩ · · · ∩Hk−j and F ∩Hk−j+1 ∩ · · · ∩Hd−j 6= 0

if and only ifC ∩H1 ∩ · · · ∩Hd−j 6= 0,

and it can have at most one such face. Using this and (40), we obtain

I =

∫G(d,d−1)n−k+j

∑C∈Fd(ηn〈k−j+1,...,d−j〉)C∩H1∩···∩Hd−j 6=0

Λr(C) νn−k+jd−1 (d(Hk−j+1, . . . ,Hn))

=d∑

p=maxr,d−k+j

2d−p∑

1≤i1<···<id−p≤k−j

∫G(d,d−1)n−k+j

×∑

Q∈Fd(Hd−j+1,...,Hn)

Q∩H1∩···∩Hd−j 6=0

Λr(Q ∩Hi1 ∩ · · · ∩Hid−p) νn−k+jd−1 (d(Hk−j+1, . . . ,Hn)).

21

We conclude that

EΛr(D[k,j]n )

=

(n−d+kk−j

)2k−jC(n− d+ k, k, j)

d∑p=maxr,d−k+j

2d−p∑

1≤i1<···<id−p≤k−j

×∫G(d,d−1)n

∑Q∈Fd(Hd−j+1,...,Hn)

Q∩H1∩···∩Hd−j 6=0

Λr(Q ∩Hi1 ∩ · · · ∩Hid−p) νnd−1(d(H1, . . . ,Hn))

=

(n−d+kk−j

)2k−jC(n− d+ k, k, j)

d∑p=maxr,d−k+j

2d−p(k − jd− p

)∫G(d,d−1)d−p

×∫G(d,d−1)n−d+p

∑Q∈Fd(Hd−j+1,...,Hn)

Q∩H1∩···∩Hd−j 6=0

Λr(Q ∩H1 ∩ · · · ∩Hd−p)

× νn−d+pd−1 (d(Hd−p+1, . . . ,Hn)) νd−pd−1(d(H1, . . . ,Hd−p)).

To evaluate the inner integral above, we fix H1, . . . ,Hd−p in general position and writeH1 ∩ · · · ∩ Hd−p =: Lp. The image measure of νd−1 under the (νd−1 almost everywherewell defined) map H 7→ H ∩ Lp from G(d, d − 1) to the Grassmannian G(Lp, p − 1) of(p−1)-dimensional subspaces of Lp is the invariant probability measure µp−1 on G(Lp, p−1).Therefore, the inner integral can be written as∫

G(d,d−1)n−d+p

∑Q∈Fd(Hd−j+1,...,Hn)

Q∩H1∩···∩Hd−j 6=0

Λr(Q ∩ Lp) νn−d+pd−1 (d(Hd−p+1, . . . ,Hn))

=

∫G(Lp,p−1)n−d+p

∑C∈Fp(hd−j+1,...,hn)

C∩hd−p+1∩···∩hd−j 6=0

Λr(C)µn−d+pp−1 (d(hd−p+1, . . . , hn)).

Note that p ≥ j. If p = j, then the second condition under the last sum is empty and can bedeleted. Here Fp(hd−j+1, . . . , hn) denotes the set of Schlafli cones in Lp that are generatedby the (p− 1)-planes hd−j+1, . . . , hn in Lp. Identifying Lp with Rp, we can apply (38) in Lp.For this, we replace d by p, the number n by n−d+p, and raise the indices of the integrationvariables in (38) by d− p. Then (38), with g = Λr, reads∫

G(Lp,p−1)n−d+p

∑C∈Fp(hd−j+1,...,hn)

C∩hd−p+1∩···∩hd−j 6=0

Λr(C)µn−d+pp−1 (d(hd−p+1, . . . , hn))

= 2C(n− d+ j, p)E(ΛrUp−j)(S(p)n−d+j),

where S(p)m denotes the (µp−1,m)- Schlafli cone in Lp. We conclude that

EΛr(D[k,j]n ) =

2(n−d+kk−j

)2k−jC(n− d+ k, k, j)

d∑p=maxr,d−k+j

2d−p(k − jd− p

)

× C(n− d+ j, p)E(ΛrUp−j)(S(p)n−d+j). (42)

22

Comparing (42) and (37), we arrive at

E(ΛrYd−k+j,d−k)(Sn−d+k)

=

(n−d+kk−j

)C(n− d+ k, d)

d∑p=maxr,d−k+j

2d−p(k − jd− p

)C(n− d+ j, p)E(ΛrUp−j)(S

(p)n−d+j).

Here we substitute d− k+ j = s and d− k = t. Then we replace n by n+ t and assume thatn > d− s. The result is

E(Ys,tΛr)(Sn)

=

(nd−s)

C(n, d)

d∑p=maxr,s

2d−p(d− sd− p

)C(n− d+ s, p)E(Up−s+tΛr)(S

(p)n−d+s).

This is the conical (or spherical) counterpart to [13, Thm. 11.7.1]. (The result is also truefor n < d− s, since then both sides of the equation are zero.)

We specialize the latter to t = s− 1. We have Ys,s−1 = Λs. Further, Up−s+t = Up−1 = Vp

in a space of dimension p. The value of E(VpΛr)(S(p)n−d+s) is seen from (35). In this way, we

obtain the following result.

Theorem 8.1. The face contents of the (νd−1, n)-Schlafli cone Sn satisfy

E(ΛsΛr)(Sn) =

(nd−s)

C(n, d)

d∑p=maxr,s

2d−p(d− sd− p

)(n− d+ s

p− r

)θ(n− d− p+ r + s, p) (43)

for r, s = 1, . . . , d, where θ is defined by (32).

This is the conical counterpart to [13, Corollary to Thm. 11.7.1].

Theorem 8.1 holds for all n ∈ N. In fact, the right side of (43) is symmetric in r ands, and if n < d − r (or n < d − s), then both sides of (43) are zero. For n = d − r (orn = d− s) equation (43) is equivalent to (26). Also note that (35) is obtained as the specialcase s = d− k and r = d of (43).

Since the expectations EΛr(Sn) are known by (26), Theorem 8.1 allows us to write downthe complete covariance matrix for the random vector (Λ1(Sn), . . . ,Λd(Sn)).

For the Cover–Efron cone Cn, there is only one second moment that we can obtain fromTheorem 8.1 by dualization, namely Ef2d−1(Cn) = Ef21 (Sn) = 4EΛ2

1(Sn) for n ≥ d.

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Authors’ addresses:

Daniel HugKarlsruhe Institute of TechnologyDepartment of MathematicsD-76128 Karlsruhe, Germanye-mail: daniel.hug@kit.edu

Rolf SchneiderAlbert-Ludwigs-UniversitatMathematisches InstitutD-79104 Freiburg i. Br., Germanye-mail: rolf.schneider@math.uni-freiburg.de

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